MARK KAC SEMINAR

We exhibit classes of interacting particle systems in discrete and continuous time, with the somewhat surprising property of having a unique translation-invariant and time-stationary measure, which is not the long-time limit of all starting measures.

This is related to an old question stated for example in the book of Liggett and examples of this phenomenon have been recently given by Maes and Shlosman for a continuous-spin model driven by Brownian Motion (Langevin dynamics) as well as by Chassaing and Mairesse for a system with synchronous updating with local translation-invariant interactions (probabilistic cellular automata). Our construction of a transition mechanism builds on the three dimensional lattice rotator model in the phase transition region under local discretization to create macroscopically coherent time periodic behavior. Proofs also rely on relative entropy techniques.

We study a branching Brownian motion Z in R^d, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the process when hit by a particle. Considering a general offspring distribution, we find the decay rate of the annealed probability that none of the particles of Z hits a trap, asymptotically in time. This problem motivates the proof of a general result about the speed of branching Brownian motion conditioned on non-extinction. In the proof, an appropriate skeleton decomposition for the underlying Galton-Watson process is provided and it is shown through a non-trivial comparison that the particles of finite line of descent do not contribute to the asymptotic decay rate of the annealed survival probability.

(this is joint work with Mine Çağlar and János Engländer)